They are both similar. Presume's definition states that it is to assume. However, the definition of presume says that it is to assume something is true in the absence of proof to the contrary, while assuming is to take for granted without proof.

They are both very similar. However, in my experience, presume has a connotation along the lines of being out of reach or above one's station. For example. I might assume that it would be ok to hug my wife's best friend the first time I met her, but I would never presume it would be ok to hug the Queen of England.